## Towards the Notion of Stability of Approximation for Hard Optimization Tasks and the Traveling Salesman Problem (1999)

Venue: | Comput. Sci |

Citations: | 14 - 2 self |

### BibTeX

@ARTICLE{Böckenhauer99towardsthe,

author = {Hans-Joachim Böckenhauer and Juraj Hromkovic and Ralf Klasing and Sebastian Seibert and Walter Unger},

title = {Towards the Notion of Stability of Approximation for Hard Optimization Tasks and the Traveling Salesman Problem},

journal = {Comput. Sci},

year = {1999},

volume = {285},

pages = {3--24}

}

### OpenURL

### Abstract

The investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory. The aim of this paper is twofold. First, we introduce the notion of stability of approximation algorithms. This notion is shown to be of practical as well as of theoretical importance, especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit polynomial-time approximation. Secondly, we apply our concept to the study of the traveling salesman problem. We show how to modify the Christofides algorithm for \Delta-TSP to obtain efficient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Ba...

### Citations

10958 |
Computers and Intractability: A Guide to the Theory of NP-completeness
- Garey, Johnson
- 1990
(Show Context)
Citation Context ...ty of Approximation Algorithms We assume that the reader is familiar with the basic concepts and notions of algorithmics and complexity theory as presented in standard textbooks like [BC93], [CLR90], =-=[GJ79]-=-, [Ho96], [Pa94]. Next, we give a new definition of the notion of an optimization problem. The reason to do this is to obtain the possibility to study the influence of the input sets on the hardness o... |

680 |
Approximation Algorithms For Combinatorial Problems
- Johnson
- 1974
(Show Context)
Citation Context ...an optimization problem does not admit an efficiently computable optimal solution, is there a possibility to efficiently compute at least an approximation of the optimal solution? Several researchers =-=[Jo74]-=-, [Lo75], [Chr76], [IK75] provided already in the middle of the seventies a positive answer for some optimization problems. It is a fascinating effect if one can jump from exponential complexity (a hu... |

645 | Some optimal inapproximability results - Hastad - 1997 |

320 | Polynomial time approximation schemes for Euclidean TSP and other geometric problems.Proc
- Arora
- 1996
(Show Context)
Citation Context ... distances between the nodes correspond to the distances in the Euclidean metrics. The Euclidean TSP is NP-hard [Pa77], but for every ff ? 1 one can design a polynomial-time ffapproximation algorithm =-=[Ar96]-=-, [Mi96]. Moreover, if one allows randomization the resulting approximation algorithm works in n \Delta (log 2 n) O(1) time [Ar97]. 2 This is the reason why we propose again to revise the notion of tr... |

273 |
Worst-Case Analysis of a New Heuristic for the Traveling Salesman Problem
- CHRISTOFIDES
- 1976
(Show Context)
Citation Context ...problem does not admit an efficiently computable optimal solution, is there a possibility to efficiently compute at least an approximation of the optimal solution? Several researchers [Jo74], [Lo75], =-=[Chr76]-=-, [IK75] provided already in the middle of the seventies a positive answer for some optimization problems. It is a fascinating effect if one can jump from exponential complexity (a huge inevitable amo... |

187 |
Fast approximation algorithms for the knapsack and sum of subset problems
- Ibarra, Kim
- 1975
(Show Context)
Citation Context ...oes not admit an efficiently computable optimal solution, is there a possibility to efficiently compute at least an approximation of the optimal solution? Several researchers [Jo74], [Lo75], [Chr76], =-=[IK75]-=- provided already in the middle of the seventies a positive answer for some optimization problems. It is a fascinating effect if one can jump from exponential complexity (a huge inevitable amount of p... |

164 | Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems
- Mitchell
- 1999
(Show Context)
Citation Context ...es between the nodes correspond to the distances in the Euclidean metrics. The Euclidean TSP is NP-hard [Pa77], but for every ff ? 1 one can design a polynomial-time ffapproximation algorithm [Ar96], =-=[Mi96]-=-. Moreover, if one allows randomization the resulting approximation algorithm works in n \Delta (log 2 n) O(1) time [Ar97]. 2 This is the reason why we propose again to revise the notion of tractabili... |

143 | The Traveling Salesman Problem With Distances One and Two - Papadimitriou, Yannakakis - 1993 |

90 | Nearly linear time approximation schemes for Euclidean TSP and other geometric problems.Proc. 38th Symposium on Foundations of Computer Science
- Arora
- 1997
(Show Context)
Citation Context ...ery ff ? 1 one can design a polynomial-time ffapproximation algorithm [Ar96], [Mi96]. Moreover, if one allows randomization the resulting approximation algorithm works in n \Delta (log 2 n) O(1) time =-=[Ar97]-=-. 2 This is the reason why we propose again to revise the notion of tractability especially because of the standard definition of complexity as the worst-case complexity: Our aim is to try to separate... |

74 |
The euclidean traveling salesman problem is NPcomplete
- Papadimitriou
- 1977
(Show Context)
Citation Context ...6]. The situation is even more interesting if one considers the Euclidean TSP, where the distances between the nodes correspond to the distances in the Euclidean metrics. The Euclidean TSP is NP-hard =-=[Pa77]-=-, but for every ff ? 1 one can design a polynomial-time ffapproximation algorithm [Ar96], [Mi96]. Moreover, if one allows randomization the resulting approximation algorithm works in n \Delta (log 2 n... |

69 |
Leiserson C, Rivest R, “Introduction to Algorithms
- Cormen
- 1998
(Show Context)
Citation Context ...e Stability of Approximation Algorithms We assume that the reader is familiar with the basic concepts and notions of algorithmics and complexity theory as presented in standard textbooks like [BC93], =-=[CLR90]-=-, [GJ79], [Ho96], [Pa94]. Next, we give a new definition of the notion of an optimization problem. The reason to do this is to obtain the possibility to study the influence of the input sets on the ha... |

67 |
A compendium of NP optimization problems. http://www.nada.kth.se/ viggo/problemlist
- Crescenzi, Kann
(Show Context)
Citation Context ...)jx 2 L I " \Sigma n I g from N to N, where TimeA (x) is the length of the computation of A on x. 2 Next, we give the definitions of standard notions in the area of approximation algorithms (see =-=e.g. [CK98]-=-, [Ho96]). Definition 3 Let U = (\Sigma I ; \Sigma O ; L; L I ; M; cost; goal) be an optimization problem, and let A be a consistent algorithm for U . For every x 2 L I , the approximation ratio RA (x... |

23 | An explicit lower bound for TSP with distances one and two - Engebretsen - 2003 |

16 |
Hochbaum (Ed.), Approximation Algorithms for NP-Hard Problems
- S
- 1997
(Show Context)
Citation Context ...proximation Algorithms We assume that the reader is familiar with the basic concepts and notions of algorithmics and complexity theory as presented in standard textbooks like [BC93], [CLR90], [GJ79], =-=[Ho96]-=-, [Pa94]. Next, we give a new definition of the notion of an optimization problem. The reason to do this is to obtain the possibility to study the influence of the input sets on the hardness of the pr... |

11 |
Performance guarantees for approximation algorithms depending on parametrized triangle inequalities
- Andreae, Bandelt
- 1995
(Show Context)
Citation Context ...ation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Bandelt =-=[AB95]-=-. Keywords: Stability of approximation, Traveling Salesman Problem 1 Introduction Immediately after introducing NP-hardness (completeness) [Co71] as a concept for proving intractability of computing p... |

8 | Performance guarantees for the TSP with a parameterized triangle inequality
- Bender, Chekuri
- 2000
(Show Context)
Citation Context ...riginal 2-approximation algorithm for \Delta-TSP. Note that, after this paper was written, we got the information about the independent, unpublished result of Bender and Chekuri, accepted for WADS'99 =-=[BC99]-=-. They designed a 4fi-approximation algorithm which can be seen as a modification of the 2-approximation algorithm for \Delta fi -TSP. Despite this nice result, there are three reasons to consider our... |

6 |
eds.): The Travelling Salesman Problem
- Lawler, Lenstra, et al.
- 1985
(Show Context)
Citation Context ...ven in the last section. In the next section we show a transparent application of our concept for TSP. 3 Stability of Approximation Algorithms and TSP We consider the well-known TSP problem (see e.g. =-=[LLRS85]-=-) that is in its general form very hard for approximation. But if one considers complete graphs in which the triangle inequality holds, then we have a 3 2 -approximation algorithm due to Christofides ... |

5 |
Cook: The complexity of theorem proving procedures
- A
- 1971
(Show Context)
Citation Context ...actor. This improves the result of Andreae and Bandelt [AB95]. Keywords: Stability of approximation, Traveling Salesman Problem 1 Introduction Immediately after introducing NP-hardness (completeness) =-=[Co71]-=- as a concept for proving intractability of computing problems, the following question has been posed: If an optimization problem does not admit an efficiently computable optimal solution, is there a ... |

2 | Crescenzi: Introduction to the Theory - Bovet, P - 1993 |

1 | Stability of approximation algorithms and the knapsack problem. Unpublished manuscript - Hromkovic - 1998 |

1 |
On the ratio of the optimal integral and functional covers
- Lov'asz
- 1975
(Show Context)
Citation Context ...ization problem does not admit an efficiently computable optimal solution, is there a possibility to efficiently compute at least an approximation of the optimal solution? Several researchers [Jo74], =-=[Lo75]-=-, [Chr76], [IK75] provided already in the middle of the seventies a positive answer for some optimization problems. It is a fascinating effect if one can jump from exponential complexity (a huge inevi... |